In this paper a new class of fuzzy sets, namely r-(τi, τj)-θ-generalized fuzzy closed sets is introduced for smooth bitopological spaces. This class falls strictly in between the class of r-(τj, τi)fuzzy θ-closed sets and the class of r-(τi, τj)generalized fuzzy closed sets. Furthermore, by using the class of r-(τi, τj)-θgeneralized fuzzy closed sets we establish a new fuzzy closure operator wh...

In this paper, we prove that any subreduct of the class of representable relation algebras whose similarity type includes intersection, relation composition and converse is a non-finitely axiomatizable quasivariety and that its equational theory is not finitely based. We show the same result for subreducts of the class of representable cylindric algebras of dimension at least three whose simila...

Several new formulations of the notion of cylindric algebra are presented. The class CA of all cylindric algebras of degree a is shown to be definitionally equivalent to a class of algebras in which only substitutions (together with the Boolean +, •, and — ) are taken to be primitive operations. Then CA is shown to be definitionally equivalent to an equational class of algebras in which only su...

The four colour theorem states that the vertices of every planar graph can be coloured with at most four colours so that no two adjacent vertices receive the same colour. This theorem is famous for many reasons, including the fact that its original 1977 proof includes a non-trivial computer verification. Recently, a formal proof of the theorem was obtained with the equational logic program Coq ...

Abstract A logic is said to admit an equational completeness theorem when it can be interpreted into the consequence relative some class of algebras. We characterize logics admitting that are either locally tabular or have tautology. In particular, shown a protoalgebraic admits precisely has two distinct logically equivalent formulas. While problem determining whether decidable both for present...

A Leibniz class is a of logics closed under the formation term-equivalent logics, compatible expansions, and non-indexed products sets logics. We study complete lattice all classes, called hierarchy. In particular, it proved that classes truth-equational assertional are meet-prime in hierarchy, while protoalgebraic equivalential meet-reducible. However, last two shown to be determined by condit...

A special algebra called EQ-algebra has been recently introduced by Vilém Novák. Its original motivation comes from fuzzy type theory, in which the main connective is fuzzy equality. EQ-algebras have three binary operations meet, multiplication, fuzzy equality and a unit element. They open the door to an alternative development of fuzzy (manyvalued) logic with the basic connective being fuzzy e...

Ehresmann semigroups may be viewed as biunary equipped with domain and range operations satisfying some equational laws. Motivated by of the main examples, we here define ordered semigroups, consider their basic properties well special cases in which order is algebraically definable. In particular, one two-sided restriction natural orders are characterized within class semigroups. The result an...

Suppose that T is an equational theory of groups or of rings. If T is finitely axiomatizable, then there is a least number μ so that T can be axiomatized by μ equations. This μ can depend on the operation symbols that occur in T . In the 1960’s, Alfred Tarski and Thomas C. Green completely determined the values of μ for arbitrary equational theories of groups and of rings. While Tarski and Gree...

In less than a decade, Deduction with Constraints (DwC) has opened a new universe in computer science. DwC can be viewed from two perspectives: one related to the automated deduction framework 17, 19], the other to the development and usage of programming languages based on logic 7, 14]. Within the automated deduction framework, constraints on the generic data structure \terms" have become a po...